Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
890 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 890 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
2^4,3,2^4|a^2=d,b^a=bc,e^a=e^2,f^a=fg,h^a=hi,f^b=fh,g^b=gi,f^e=g,g^e=fg,h^e=i,i^e=hi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72220367, #72220578, #72220591, … |
| maximal normal subgroup of | #72220367, #72220578, #72220591, … |
| maximal quotient of | #72220365, #72238101, #72238102, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085713);